Remainder Theorem

Fast method · No long division

Remainder Theorem

Divide by x − a → substitute x = a → remainder = P(a).

x − a→x = a→P(a)

Core concept

The theorem

If a polynomial P(x) is divided by (x − a), then the remainder is:

Remainder = P(a)

Identify the divisor

Write it in the form x − a.

Find a

Read the sign carefully.

Substitute

Evaluate P(a).

Interpret

If P(a)=0, then x−a is a factor.

Method

Use the theorem correctly

Match the form

Compare the divisor with x − a.

Determine a

For x + 3, use a = −3.

Substitute

Replace every x in the polynomial with a.

Simplify

Use brackets for negative values.

State the answer

The value of P(a) is the remainder.

x − 4a = 4

x + 3a = −3

2x − 62(x − 3), so a = 3

⚠ For x + 5, substitute x = −5, not x = 5.

Worked examples

Learn one step at a time

Example 1

Guided practice

Try it yourself

Practice question

Find the remainder when P(x)=2x³−3x²+4x−5 is divided by x−2.

Interactive tool

Remainder calculator

x⁴ x³ x² x constant a

Enter the coefficients and the value of a.

Assessment

Remainder theorem quiz

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Ready to test your understanding?

12 questions with instant, step-based feedback.

1Identify aWrite the divisor as x − a.

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2Calculate P(a)Substitute and simplify.

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3State the remainderThe value obtained is the answer.